A base station is the core component in a telecommunication system. Ideally, both the base station and a mobile terminal should transmit or receive signal using a perfect crystal whose frequency accuracy is perfectly good. Due to cost, temperature, and other affects, however, there is some bias between the crystal frequencies of the base station and the mobile terminal. Furthermore, there are Doppler effects and the mobile terminal has temperature drift. As a result, there is a frequency difference between the carrier frequency of the received signal and the frequency of local crystal. Without compensation, the frequency difference will degrade the performance of the mobile terminal greatly. So the frequency offset estimation followed by frequency offset compensation is very important in the mobile terminal in order to maintain good performance of the mobile terminal. A need exists to improve the accuracy and stability of frequency offset estimation in telecommunication systems.